Abstract

A nonlinear joint transform image correlator is investigated. The Fourier transform interference intensity is thresholded to provide higher correlation peak intensity and a better defined correlation spot. Analytical expressions for the thresholded joint power spectrum are provided. The effects of nonlinearity at the Fourier plane on the correlation signals at the output plane are investigated. The correlation signals are determined in terms of nonlinear characteristics of the spatial light modulator (SLM) at the Fourier plane. We show that thresholding the interference intensity results in a sum of infinite harmonic terms. Each harmonic term is envelope modulated due to the nonlinear characteristics of the device and phase modulated by m times the phase modulation of the nonthresholded joint power spectrum. The correct phase information about the correlation signal is recovered from the first-order harmonic of the thresholded interference intensity. We show that various types of autocorrelation signal can be produced simply by varying the severity of the nonlinearity and without the need to synthesize the specific matched filter. For example, the autocorrelation signal produced by a phase-only matched filter can be obtained by selecting the appropriate nonlinearity.

© 1989 Optical Society of America

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References

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  1. H. K. Liu, Ed., Optical and Digital Pattern Recognition, Proc. Soc. Photo-Opt. Instrum. Eng.754, (1987).
  2. H. J. Caulfield, “Role of the Horner Efficiency in the Optimization of Spatial Filters for Optical Pattern Recognition,” Appl. Opt. 21, 4391–4392 (1982).
    [CrossRef] [PubMed]
  3. J. L. Horner, H. O. Bartelt, “Two-Bit Correlation,” Appl. Opt. 24, 2889–2893 (1985).
    [CrossRef] [PubMed]
  4. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
    [CrossRef]
  5. D. Flannery et al., “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
    [CrossRef]
  6. D. A. Gregory, “Review of Compact Optical Correlators,” Proc. Soc. Photo-Opt. Instrum. Eng. 960, 66–85 (1988).
  7. H. Arsenault, “Performance of the Tandem Component Matched Filter for Pattern Recognition,” Opt. Commun. 65, 334–339 (1988).
    [CrossRef]
  8. B. Javidi, C. J. Kuo, “Joint Transform Image. Correlation Using a Binary Spatial Light Modulator at the Fourier Plane,” Appl. Opt. 27, 663–665 (1988); J. Opt. Soc. Am. A 4(13), P86 (1987).
    [CrossRef] [PubMed]
  9. B. Javidi, “Multifunction Nonlinear Optical Processor,” Opt. Eng. 28, August1989, Vol. 8 (1989).
  10. B. Javidi, J. L. Horner, “Single Spatial Light Modulator Joint Transform Correlator,” Appl. Opt. 28, 1027–1032 (1989).
    [CrossRef] [PubMed]
  11. B. Javidi, “Comparison of Binary Joint Transform Correlators and Phase-Only Matched Filter Based Correlators,” Opt. Eng. 28, 267–272 (1989).
    [CrossRef]
  12. C. S. Weaver, J. W. Goodman, “A Technique for Optically Convolving Two Functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  13. J. E. Rau, “Detection of Differences in Real Distributions,” J. Opt. Soc. Am. 56, 1490–1494 (1966).
    [CrossRef]
  14. W. R. Bennett, “Spectra of Quantized Signals,” Bell Syst. Tech. J. 27, 446–451 (1948).
  15. D. Middleton, “Some General Results in the Theory of Noise Through Nonlinear Devices,” Q. Appl. Math. 5, 445–498 (1948).
  16. A. Kozma, D. L. Kelly, “Spatial Filtering for Detection of Signals Submerged in Noise,” Appl. Opt. 4, 387–392 (1965).
    [CrossRef]
  17. A. Kozma, “Photographic Recording of Spatially Modulated Coherent Light,” J. Opt. Soc. Am. 56, 428–432 (1966).
    [CrossRef]
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1967).

1989

B. Javidi, “Multifunction Nonlinear Optical Processor,” Opt. Eng. 28, August1989, Vol. 8 (1989).

B. Javidi, J. L. Horner, “Single Spatial Light Modulator Joint Transform Correlator,” Appl. Opt. 28, 1027–1032 (1989).
[CrossRef] [PubMed]

B. Javidi, “Comparison of Binary Joint Transform Correlators and Phase-Only Matched Filter Based Correlators,” Opt. Eng. 28, 267–272 (1989).
[CrossRef]

1988

D. Flannery et al., “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
[CrossRef]

D. A. Gregory, “Review of Compact Optical Correlators,” Proc. Soc. Photo-Opt. Instrum. Eng. 960, 66–85 (1988).

H. Arsenault, “Performance of the Tandem Component Matched Filter for Pattern Recognition,” Opt. Commun. 65, 334–339 (1988).
[CrossRef]

B. Javidi, C. J. Kuo, “Joint Transform Image. Correlation Using a Binary Spatial Light Modulator at the Fourier Plane,” Appl. Opt. 27, 663–665 (1988); J. Opt. Soc. Am. A 4(13), P86 (1987).
[CrossRef] [PubMed]

1985

1984

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

1982

1966

1965

A. Kozma, D. L. Kelly, “Spatial Filtering for Detection of Signals Submerged in Noise,” Appl. Opt. 4, 387–392 (1965).
[CrossRef]

1948

W. R. Bennett, “Spectra of Quantized Signals,” Bell Syst. Tech. J. 27, 446–451 (1948).

D. Middleton, “Some General Results in the Theory of Noise Through Nonlinear Devices,” Q. Appl. Math. 5, 445–498 (1948).

Arsenault, H.

H. Arsenault, “Performance of the Tandem Component Matched Filter for Pattern Recognition,” Opt. Commun. 65, 334–339 (1988).
[CrossRef]

Bartelt, H. O.

Bennett, W. R.

W. R. Bennett, “Spectra of Quantized Signals,” Bell Syst. Tech. J. 27, 446–451 (1948).

Caulfield, H. J.

Flannery, D.

D. Flannery et al., “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
[CrossRef]

Goodman, J. W.

Gregory, D. A.

D. A. Gregory, “Review of Compact Optical Correlators,” Proc. Soc. Photo-Opt. Instrum. Eng. 960, 66–85 (1988).

Horner, J. L.

Javidi, B.

B. Javidi, “Comparison of Binary Joint Transform Correlators and Phase-Only Matched Filter Based Correlators,” Opt. Eng. 28, 267–272 (1989).
[CrossRef]

B. Javidi, “Multifunction Nonlinear Optical Processor,” Opt. Eng. 28, August1989, Vol. 8 (1989).

B. Javidi, J. L. Horner, “Single Spatial Light Modulator Joint Transform Correlator,” Appl. Opt. 28, 1027–1032 (1989).
[CrossRef] [PubMed]

B. Javidi, C. J. Kuo, “Joint Transform Image. Correlation Using a Binary Spatial Light Modulator at the Fourier Plane,” Appl. Opt. 27, 663–665 (1988); J. Opt. Soc. Am. A 4(13), P86 (1987).
[CrossRef] [PubMed]

Kelly, D. L.

A. Kozma, D. L. Kelly, “Spatial Filtering for Detection of Signals Submerged in Noise,” Appl. Opt. 4, 387–392 (1965).
[CrossRef]

Kozma, A.

A. Kozma, “Photographic Recording of Spatially Modulated Coherent Light,” J. Opt. Soc. Am. 56, 428–432 (1966).
[CrossRef]

A. Kozma, D. L. Kelly, “Spatial Filtering for Detection of Signals Submerged in Noise,” Appl. Opt. 4, 387–392 (1965).
[CrossRef]

Kuo, C. J.

Middleton, D.

D. Middleton, “Some General Results in the Theory of Noise Through Nonlinear Devices,” Q. Appl. Math. 5, 445–498 (1948).

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Rau, J. E.

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Weaver, C. S.

Appl. Opt.

A. Kozma, D. L. Kelly, “Spatial Filtering for Detection of Signals Submerged in Noise,” Appl. Opt. 4, 387–392 (1965).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

W. R. Bennett, “Spectra of Quantized Signals,” Bell Syst. Tech. J. 27, 446–451 (1948).

J. Opt. Soc. Am.

A. Kozma, “Photographic Recording of Spatially Modulated Coherent Light,” J. Opt. Soc. Am. 56, 428–432 (1966).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng.

D. Flannery et al., “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
[CrossRef]

Opt. Commun.

H. Arsenault, “Performance of the Tandem Component Matched Filter for Pattern Recognition,” Opt. Commun. 65, 334–339 (1988).
[CrossRef]

Opt. Eng.

B. Javidi, “Multifunction Nonlinear Optical Processor,” Opt. Eng. 28, August1989, Vol. 8 (1989).

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

B. Javidi, “Comparison of Binary Joint Transform Correlators and Phase-Only Matched Filter Based Correlators,” Opt. Eng. 28, 267–272 (1989).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

D. A. Gregory, “Review of Compact Optical Correlators,” Proc. Soc. Photo-Opt. Instrum. Eng. 960, 66–85 (1988).

Q. Appl. Math.

D. Middleton, “Some General Results in the Theory of Noise Through Nonlinear Devices,” Q. Appl. Math. 5, 445–498 (1948).

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1967).

H. K. Liu, Ed., Optical and Digital Pattern Recognition, Proc. Soc. Photo-Opt. Instrum. Eng.754, (1987).

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Figures (5)

Fig. 1
Fig. 1

Nonlinear optical correlator: (a) implementation using a variable contrast optically addressed SLM at the Fourier plane and (b) implementation using a CCD interfaced with a thresholding network and an electrically addressed SLM at the Fourier plane.

Fig. 2
Fig. 2

Thresholding network using a kth law device nonlinearity.

Fig. 3
Fig. 3

Image used in the correlation tests.

Fig. 4
Fig. 4

Thresholded modified joint power spectrum using a kth law device. k = 1 corresponds to a linear correlation and the severity of the nonlinearity increases as k decreases: (a) k = 1, (b) k = 0.9, (c) k = 0.7, (d) k = 0.5.

Fig. 5
Fig. 5

Correlation signals obtained by the thresholded modified joint power spectrum using a kth law device; k = 1 corresponds to a linear correlation and the severity of the nonlinearity increased as k decreases: (a) k = 1, (b) k = 0.9, (c) k = 0.7, (d) k = 0.5.

Tables (2)

Tables Icon

Table I Comparison Between the Nonlinear Correlation Results Obtained by the Fourier Series Expansion Technique [Eq. (42)]a and the Results Obtained by the Numerical Technique

Tables Icon

Table II Correlation Resultsa

Equations (48)

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I ( α , β ) = S ( α , β ) exp [ i φ S ( α , β ) ] exp ( i x 0 α ) + R ( α , β ) exp [ i φ R ( α , β ) ] exp ( i x 0 α ) ,
E ( α , β ) = | I ( α , β ) | 2 = S 2 ( α , β ) + R 2 ( α , β ) + S ( α , β ) × exp [ i φ S ( α , β ) ] R ( α , β ) exp [ i φ R ( α , β ) exp ( i 2 x 0 α ) + S ( α , β ) exp [ i φ S ( α , β ) ] R ( α , β ) × exp [ i φ R ( α , β ) exp ( i 2 x 0 α ) .
h ( x , y ) = R 11 ( x , y ) + R 22 ( x , y ) + R 12 ( x 2 x 0 , y ) + R 21 ( x + 2 x 0 , y ) ,
R 21 ( x , y ) = R 12 ( x , y ) = s ( ξ,ζ ) r ( ξ x , ζ y ) d ξ d ζ ,
R 11 ( x , y ) = s ( ξ,ζ ) s ( ξ x , ζ y ) d ξ d ζ , and
R 22 ( x , y ) = r ( ξ,ζ ) r ( ξ x , ζ y ) d ξ d ζ ,
G ( ω ) = g ( E ) exp ( i ω E ) d E .
g ( E ) = 1 2 π G ( ω ) exp ( i ω E ) d ω .
g ( E ) = 1 2 π G ( ω ) exp { i ω [ R 2 ( α , β ) + S 2 ( α , β ) ] } × exp { i 2 ω R ( α , β ) S ( α , β ) × cos [ 2 x 0 α + φ S ( α , β ) φ R ( α , β ) ] } d ω .
exp { i 2 ω R ( α , β ) S ( α , β ) cos [ 2 x 0 α φ R ( α , β ) + φ S ( α , β ) ] } = υ = 0 υ ( i ) υ J υ [ 2 ω R ( α , β ) S ( α , β ) ] cos [ 2 υ x 0 α + υ φ S ( α , β ) υ φ R ( α , β ) ] ,
υ = { 1 , υ = 0 , 2 , υ > 0 ,
g ( E ) = υ = 0 υ 2 π ( i ) υ G ( ω ) exp { i ω [ R 2 ( α , β ) + S 2 ( α , β ) ] } J υ [ 2 ω R ( α , β ) S ( α , β ) ] × cos [ 2 υ x 0 α + υ φ S ( α , β ) υ φ R ( α , β ) ] d ω .
g ( E ) = υ = 0 H υ [ R ( α , β ) , S ( α , β ) ] × cos [ 2 υ x 0 α + υ φ S ( α , β ) υ φ R ( α , β ) ] ,
H υ [ R ( α , β ) , S ( α , β ) ] = υ 2 π ( i ) υ G ( ω ) exp { i ω [ R 2 ( α , β ) + S 2 ( α , β ) ] } J υ [ 2 ω R ( α , β ) S ( α , β ) ] d ω .
G ( ω ) = 2 ( i ω ) k + 1 Γ ( k + 1 ) , k 1 ,
H υ [ R ( α , β ) , S ( α , β ) ] = Γ ( k + 1 ) υ π ( i ) υ k 1 1 ω k + 1 × J υ [ 2 ω R ( α , β ) s ( α , β ) ] d ω .
H υ [ R ( α , β ) , S ( α , β ) ] = 2 Γ ( k + 1 ) υ [ 2 R ( α , β ) S ( α , β ) ] k 2 k + 1 Γ ( 1 υ k 2 ) Γ ( 1 + υ + k 2 ) ,
g k ( E ) = υ = 1 ( υ , odd ) υ Γ ( k + 1 ) [ R ( α , β ) S ( α , β ) ] k 2 K Γ ( 1 υ k 2 ) Γ ( 1 + υ + k 2 ) × cos [ 2 υ x 0 α + υ φ S ( α , β ) υ φ R ( α , β ) ] .
g 1 k ( E ) = 2 Γ ( k + 1 ) [ R ( α , β ) S ( α , β ) ] k Γ ( 1 1 k 2 ) Γ ( 1 + 1 + k 2 ) × cos [ 2 x 0 α + φ S ( α , β ) φ R ( α , β ) ] .
g 1 k ( E ) = 2 Γ ( k + 1 ) Γ ( 1 1 k 2 ) Γ ( 1 + 1 + k 2 ) [ R ( α , β ) ] 2 k cos [ 2 x 0 α ] .
g 1 1 / 2 ( E ) = ( π / 2 ) [ R ( α , β ) S ( α , β ) ] 1 / 2 cos ( 2 x 0 α ) .
g 1 1 / 2 ( E ) = ( π / 2 ) R ( α , β ) cos ( 2 x 0 α ) .
H ϕ ( α , β ) = exp [ i ϕ R ( α , β ) ] ,
T ( α , β ) = S ( α , β ) exp { i [ ϕ S ( α , β ) ϕ R ( α , β ) ] } ,
T a ( α , β ) = R ( α , β ) ,
A ( α , β ) = [ R ( α , β ) / S ( α , β ) ] 1 / 2 .
g 0 ( E ) = υ = 1 υ , odd υ Γ ( 1 υ 2 ) Γ ( 1 + υ 2 ) × cos [ 2 υ x 0 α + υ φ S ( α , β ) υ φ R ( α , β ) ] .
g 10 ( E ) = 4 π cos [ 2 x 0 a + φ S ( α , β ) φ R ( α , β ) ] .
g 10 ( E ) = 4 π cos ( 2 x 0 α ) .
g ( α , β ) = { 1 , R 2 ( α , β ) + S 2 ( α , β ) + 2 R ( α , β ) S ( α , β ) cos [ 2 x 0 α + φ S ( α , β ) φ R ( α , β ) ] V T , 0 , R 2 ( α , β ) + S 2 ( α , β ) + 2 R ( α , β ) S ( α , β ) cos [ 2 x 0 α + φ S ( α , β ) φ R ( α , β ) ] < V T ,
R 2 ( α , β ) + S 2 ( α , β ) + 2 R ( α , β ) S ( α , β ) cos [ 2 x 0 d 2 φ S ( α , β ) φ R ( α , β ) ] = V T .
d = 1 x 0 cos 1 [ R 2 ( α , β ) + S 2 ( α , β ) V T 2 R ( α , β ) S ( α , β ) ] for | R 2 ( α , β ) + S 2 ( α , β ) V T 2 R ( α , β ) S ( α , β ) | 1 .
[ R ( α , β ) + S ( α , β ) ] 2 V T ,
[ R ( α , β ) S ( α , β ) ] 2 V T .
g ( α , β ) = g ( α ) = { υ = K υ exp ( i 2 υ α x 0 ) , | R 2 ( α , β ) + S 2 ( α , β ) V T 2 R ( α , β ) S ( α , β ) | 1 , 0 , else
K υ = 2 x 0 1 4 x 0 1 4 x 0 g ( α ) exp ( i 2 x 0 υ α ) d α .
K υ = { 2 x 0 d , υ = 0 , 1 π υ exp { i υ [ φ S ( α , β ) φ R ( α , β ) ] } sin ( υ x 0 d ) , υ 0 ,
g ( α , β ) = 2 x 0 d + { υ = 1 H υ [ R ( α , β ) , S ( α , β ) ] cos [ 2 υ α x 0 + υ φ s ( α , β ) υ φ R ( α , β ) ] , | R 2 ( α , β ) + S 2 ( α , β ) V T 2 R ( α , β ) S ( α , β ) | V T , 0 , else
H υ [ R ( α , β ) S ( α , β ) ] = 2 π υ sin { υ cos 1 [ R 2 ( α , β ) + S 2 ( α , β ) V T 2 R ( α , β ) S ( α , β ) ] } .
g 1 c ( α , β ) = 2 π 1 [ R 2 ( α , β ) + S 2 ( α , β ) V T 2 R ( α , β ) S ( α , β ) ] 2 × cos [ 2 x 0 α + φ S ( α , β ) φ R ( α , β ) ] ,
sin ( cos 1 x ) = 1 x 2 ,
g 2 c ( α , β ) = 1 π sin { 2 cos 1 [ R 2 ( α , β ) + S 2 ( α , β ) V T 2 R ( α , β ) S ( α , β ) ] } × cos [ 4 x 0 α + 2 φ S ( α , β ) 2 φ R ( α , β ) ] ,
g 2 c ( α , β ) = 2 π [ R 2 ( α , β ) + S 2 ( α , β ) V T 2 R ( α , β ) S ( α , β ) ] × 1 [ R 2 ( α , β ) + S 2 ( α , β ) V T 2 R ( α , β ) S ( α , β ) ] 2 × cos [ 4 x 0 α + 2 φ S ( α , β ) 2 φ R ( α , β ) ] .
R ( α , β ) exp [ i φ R ( α , β ) ] = S ( α , β ) exp [ i φ S ( α , β ) ] .
g a [ R ( α , β ) ] = { υ = 1 H υ a [ R ( α , β ) ] cos ( 2 υ x 0 α ) , 4 R 2 ( α , β ) V T , 0 ; else ,
H υ a [ R ( α , β ) ] = 2 π υ sin { υ cos 1 [ 2 R 2 ( α , β ) V T 2 R 2 ( α , β ) ] } .
g 1 a [ R ( α , β ) ] = { 2 V T π R ( α , β ) 1 V T 4 R 2 ( α , β ) cos ( 2 x 0 α ) , 4 R 2 ( α , β ) V T , 0 , else .
g 2 a ( α , β ) = { 2 V T π R ( α , β ) 1 V T 4 R 2 ( α , β ) [ 1 V T 2 R 2 ( α , β ) ] cos ( 4 x 0 α ) , 4 R 2 ( α , β ) V T , 0 , else .

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